A fire department in a rural county reports that its response time to fires is

Question

A fire department in a rural county reports that its response time to fires is approximately normally distributed with a mean of 22 minutes and a standard deviation of 3.9 minutes.

Use the empirical rule (68-95-99.7 rule) to find the proportion of response times between 4.2 and 29 minutes.

Calculate and interpret the 80th percentile of the distribution of response times.

What proportion of all response times for this fire department are below the national mean response time of 18.5 minutes?

Answer 1

Hi Xavier,

You may want to double-check the values of 4.2 and 29 for the first part of your question. The 68-95-99.7 rule states that for normal distributions, 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three, but I'm not sure how to get precise answers using the rule with those values, although that solution could be found with statistical software.

To the second part, the 80th percentile, this question involves the classic formula for normal distributions:

z=x-mu/sigma where:

x=value you are given

mu=mean

sigma=sd

z=z-score, explained below

Here, you were given 80%, or 0.8. This is your proportion. Go into the center of the z-score table--you can find online, but Wyzant reviews posts with links, so I can't link here-- and look for the closest value to 0.8. It's 0.7995, which corresponds to z-score 0.84. Now, you have your z. Thus:

z=0.84

x=x, the time we are looking for

mu=22

sigma=3.9

0.84=(x-22)/3.9

Solve for x; multiply both sides by 3.9

3.276=x-22

Add 22 on both sides:

x=25.276; 80% of fires are responded to within 25.276 minutes.

As for part three, this uses the same formula but:

z=unknown, our first objective

x=18.5, national average

mu=22, local average

sigma=3.9

z=(18.5-22)/3.9

z=-0.90

Return to your z-table; look up -0.9 on vertical column, 0 on horizontal row, this gives:

P=0.1841, or about 18.4%; makes sense given rural county.

I hope this helps you.